what areas of pure math are related to control and systems engineer

Question

i am an undergraduate control and system engineer undergraduate i had previous interest in mathematics now i want to study pure mathematics in conjunction with control engineering but it would be better for me if the pure math courses are related to control and systems so what are these areas that will be most related to my control engineering major also i wanted to note that my control and system major involves numerous communications courses. note: i have taken calculus differential equations courses up till now in my engineering major.

Answer 1

Pure math is everywhere in control theory. That being said, control theory is even more so filled with applied math, so the line between pure and applied math is necessarily very blurry. For example, the field of optimization is seen as mostly an applied field and is used everywhere in control theory, however as you study deeper, optimization looks more and more like pure math.

For pure math, I would recommend the following courses:

1. Real and complex analysis: Complex analysis is obviously used everywhere in classical control theory. Everything from inverse Laplace transform, Bode plot, Nyquist diagrams involves a good understanding of complex analysis. Similarly, real analysis is important to understand existence and uniqueness of solutions to a control system and generally useful in proof writing and developing mathematical maturity. Most signal in control theory is assumed to be measurable, so a bit background in measure theory also helps.

2. Functional analysis: control systems are operators that acts on signals drawn from certain function spaces, functional analysis is a must to understand more advanced topics.

3. Linear algebra: Much of linear control theory requires a good background in linear algebra. However, much of the linear algebra taught in engineering school is applied. Try to seek out linear algebra courses from the pure math department

4. Dynamical systems: nonlinear control theory rests squarely on the basis of dynamical systems. Control of limit cycles, chaos, hetero/homoclinic orbits etc. all requires understanding dynamical system.

Also depending on your interest, courses in differential geometry, differential topology, category theory, Riemannian geometry, graph theory could also be useful. However, most of these fields that I just mentioned at this stage too advanced, too undigested for control people to understand and to be useful. Most control theory people will know a bit of these areas, but not fully unless they specialized in those areas somehow. Maybe in 50 years we will see more applications of these areas.